The equations \(x-y+2 z=4\) \(3 x+y+4 z=6\) \(x+y+z=1\) have

On an \(n \times n\) chessboard, the total number of rectangles which are not squares is 350 . Then, the number of white squares on the chessboard is .......

For \(A, B\) and \(C\), if \(A+B+C=0\), then \(\sin (2 A)+\sin (2 B)+\sin (2 C)\) is equal to

If \(X\) is a poisson variable such that \(3 P(X=4)=\frac{1}{2} P(X=2)+P(X=0)\), then the mean \(X\) is

A line \(l\) meets the circle \(x^2+y^2=61\) in \(A, B\) and \(P(-5,6)\) is such that \(P A=P B=10\). Then, the equation of \(l\) is

If the arithmetic mean of the following frequency table is 50 , then
\begin{array}{|c|c|}
\hline Class & Frequency \\
\hline 0-20 & 17 \\
20-40 & f_1 \\
40-60 & 32 \\
60-80 & f_2 \\
80-100 & 19 \\
\hline & \mathbf{1 2 0} \\
\hline
\end{array}

Let \(\vec{a} \times \vec{b}=7 \hat{i}-5 \hat{j}-4 \hat{k}\) and \(\vec{a}=\hat{i}+3 \hat{j}-2 \hat{k}\). If the length of projection of \(\vec{b}\) on \(\vec{a}\) is \(\frac{8}{\sqrt{14}}\), then \(|\vec{b}|=\)

The type of bond connecting two nucleotides is

A person \(P\) speaks truth in \(75 \%\) cases and another person \(R\) in \(80 \%\) cases. Then, the probability that they likely to contradict each other in narrating the same event is

Variance of the following continuous frequency distribution is

Class Interval	\(0\)–\(4\)	\(4\)–\(8\)	\(8\)–\(12\)	\(12\)–\(16\)
Frequency	\(1\)	\(2\)	\(2\)	\(1\)

The formation of oxide ion \(\mathrm{O}^{2-}\) from oxygen atom requires an exothermic reaction, followed by an endothermic step as shown below. The process of formation of \(\mathrm{O}^{2-}\) in gas phase is unfavorable \(\left(\Delta H^{\ominus}=+\right.\) ve \()\), even though it has stable configuration of the nearest noble gas neon. This is because
\(\begin{aligned} & \mathrm{O}(g)+e^{-} \longrightarrow \mathrm{O}^{-}(g) ; \Delta H^{\ominus}=-141 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & \mathrm{O}^{-}(g)+e^{-} \longrightarrow \mathrm{O}^{2-}(g) ; \Delta H^{\ominus}=+760 \mathrm{~kJ} \mathrm{~mol}^{-1}\end{aligned}\)

The maximum value of \(x^4 y^4\) when \(a^2 x^4+b^2 y^4=c^6\) is

The ratio of de Broglie wavelengths associated with thermal neutrons at temperatures \(127^{\circ} \mathrm{C}\) and \(352^{\circ} \mathrm{C}\) is